Addition of angular momentum discovery of electron

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Addition of Angular Momentum Discovery of electron spin angular momentum means that total electron angular momentum of H-atom becomes J = L + S In classical mechanics this would be calculated trivially, J = J x J y J z = L x L y L z + S x S y S z But this requires that we know all 3 vector components of L and S simultaneously. In QM we can only know length of each angular momentum vector and one component simultaneously. We can only know sum of one component, J z = L z + S z which tells us that m j = m 𝓁 + m s P. J. Grandinetti (Chem. 4300) Magnetism, Ang. Mom., & Spin Nov 13, 2017 35 / 62
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Addition of Angular Momentum In QM we lack information needed to calculate length of total angular momentum vector, J . But we have constraints on how big or small the total can be. Largest vector sum is when both vectors point in same direction Smallest vector sum is when both point in opposite directions In other words when we add angular momentum vectors in QM we can only constrain total angular momentum as lying between these 2 limits | L + S | | J | | | | | L - | S | | | | P. J. Grandinetti (Chem. 4300) Magnetism, Ang. Mom., & Spin Nov 13, 2017 36 / 62
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Addition of Angular Momentum | L + S | | J | | | | | L - | S | | | | translates to j ( j + 1 ) | | | 𝓁 ( 𝓁 + 1 ) - s ( s + 1 ) | | | and constrains j to fall between 2 limits of | 𝓁 - s | j 𝓁 + s with only integral or half-integral values. For each j value there will be 2 j + 1 values of m j , that is, m j = - j , - j + 1 , , j - 1 , j P. J. Grandinetti (Chem. 4300) Magnetism, Ang. Mom., & Spin Nov 13, 2017 37 / 62
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Addition of Angular Momentum Example What are the total angular momenta resulting from the addition of 𝓁 = 0 and s = 1 2 ? Solution: j = | 0 - 1 2 | to | 0 + 1 2 | gives j = 1 2 Example What are the total angular momenta resulting from the addition of 𝓁 = 1 and s = 1 2 ? Solution: j = | 1 - 1 2 | to | 1 + 1 2 | gives j = 1 2 and 3 2 P. J. Grandinetti (Chem. 4300) Magnetism, Ang. Mom., & Spin Nov 13, 2017 38 / 62
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Electron Magnetic Moment from Total Angular Momentum Magnetic moment associated with electron having given value of j is 𝜇 j = g j 𝜇 B j ( j + 1 ) g j is called Landé g factor g j = j ( j + 1 )( g 𝓁 + g s ) + [ 𝓁 ( 𝓁 + 1 ) - s ( s + 1 )]( g 𝓁 - g s ) 2 j ( j + 1 ) Total magnetic dipole moment operator for e - is ̂𝜇 = - g j 𝜇 B ̂ J and for total z component of the magnetic dipole moment operator we have ̂𝜇 z = - g j 𝜇 B ̂ J z P. J. Grandinetti (Chem. 4300) Magnetism, Ang. Mom., & Spin Nov 13, 2017 39 / 62
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Clebsch-Gordon series expansion How do the wave functions associated with 𝓁 , m 𝓁 , s , and m s combine to form the wave functions associated with j and m j ? They are related through Clebsch-Gordon series expansion, ? j , m j = 𝓁 m 𝓁 =- 𝓁 s m s =- s C ( j , m j , 𝓁 , m 𝓁 , s , m s ) ? 𝓁 , m 𝓁 , s , m s C ( j , m j , 𝓁 , m 𝓁 , s , m s ) are called Clebsch-Gordon coefficients. These coefficients are nonzero only when m j = m 𝓁 + m s .
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